I have been reading "The Entrepreneurial Earnings Puzzle: Mismeasurement or Real?" a working paper by Thomas Åstebro and Jing Chen, that looks at the research literature on entrepreneurs' earnings (paper available here). See the paper for a literature review; research generally finds, among other things, that entrepreneurs earn on average less than employed workers and work longer hours, and continue to do so despite having better opportunities by being an employed worker.
With a supposed earnings disadvantage, why would anyone choose to be an entrepreneur willingly? Åstebro and Chen specifically investigate if the earnings gap between self-employed and employed workers is due to underreported income of the self-employed. They base their analysis on food consumption patterns—take two people with the same income, self-employed and employed workers, and see how their food consumption differ. They find that self-employed spend more on food than similar employed persons, suggesting an underreporting of income by the self-employed by significant degrees. This assumes that food preferences are the same across individuals (among other assumptions), so the authors conclude that this is partial, suggestive evidence of underreporting of income. Regardless, it does not account for other theories on why people choose to become entrepreneurs.
My primary takeaway about entrepreneurial earnings:
"In general, predictors of entrepreneurial earnings are weak... As researchers, we must remain humble regarding our ability to explain or predict entrepreneurial success. By far the most prominent explanation is chance."